Calculus of variations

In order to illustrate this process, consider the problem of finding the extremal function y = f (x) , which is the shortest curve that connects two points (x₁, y₁) and (x₂, y₂) . The arc length of the curve is given by

${\displaystyle A[y]=\int _{x_{1}}^{x_{2}}{\sqrt {1+[y'(x)]^{2}}}\,dx\,,}$

with

${\displaystyle y\,'(x)={\frac {dy}{dx}}\,,\ \ y_{1}=f(x_{1})\,,\ \ y_{2}=f(x_{2})\,.}$

[lower-alpha 9]

The Euler–Lagrange equation will now be used to find the extremal function f (x) that minimizes the functional A[y ] .

${\displaystyle {\frac {\partial L}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L}{\partial f'}}=0}$

with

${\displaystyle L={\sqrt {1+[f'(x)]^{2}}}\,.}$

Since f does not appear explicitly in L , the first term in the Euler–Lagrange equation vanishes for all f (x) and thus,

${\displaystyle {\frac {d}{dx}}{\frac {\partial L}{\partial f'}}=0\,.}$

Substituting for L and taking the derivative,

${\displaystyle {\frac {d}{dx}}\ {\frac {f'(x)}{\sqrt {1+[f'(x)]^{2}}}}\ =0\,.}$

Thus

${\displaystyle {\frac {f'(x)}{\sqrt {1+[f'(x)]^{2}}}}=c\,,}$

for some constant c. Then

${\displaystyle {\frac {[f'(x)]^{2}}{1+[f'(x)]^{2}}}=c^{2}\,,}$

where

${\displaystyle 0\leq c^{2}<1.}$

Solving, we get

${\displaystyle [f'(x)]^{2}={\frac {c^{2}}{1-c^{2}}}\,}$

which implies that

${\displaystyle f'(x)=m}$

is a constant and therefore that the shortest curve that connects two points (x₁, y₁) and (x₂, y₂) is

${\displaystyle f(x)=mx+b\qquad {\text{with}}\ \ m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\quad {\text{and}}\quad b={\frac {x_{2}y_{1}-x_{1}y_{2}}{x_{2}-x_{1}}}}$

and we have thus found the extremal function f(x) that minimizes the functional A[y] so that A[f] is a minimum. The equation for a straight line is y = f(x). In other words, the shortest distance between two points is a straight line.[lower-alpha 10]

It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by

${\displaystyle V[\varphi ]={\frac {1}{2}}\iint _{D}\nabla \varphi \cdot \nabla \varphi \,dx\,dy.\,}$

The functional V is to be minimized among all trial functions φ that assume prescribed values on the boundary of D. If u is the minimizing function and v is an arbitrary smooth function that vanishes on the boundary of D, then the first variation of ${\displaystyle V[u+\varepsilon v]}$ must vanish:

${\displaystyle {\frac {d}{d\varepsilon }}V[u+\varepsilon v]|_{\varepsilon =0}=\iint _{D}\nabla u\cdot \nabla v\,dx\,dy=0.\,}$

Provided that u has two derivatives, we may apply the divergence theorem to obtain

${\displaystyle \iint _{D}\nabla \cdot (v\nabla u)\,dx\,dy=\iint _{D}\nabla u\cdot \nabla v+v\nabla \cdot \nabla u\,dx\,dy=\int _{C}v{\frac {\partial u}{\partial n}}\,ds,}$

where C is the boundary of D, s is arclength along C and ${\displaystyle \partial u/\partial n}$ is the normal derivative of u on C. Since v vanishes on C and the first variation vanishes, the result is

${\displaystyle \iint _{D}v\nabla \cdot \nabla u\,dx\,dy=0\,}$

for all smooth functions v that vanish on the boundary of D. The proof for the case of one dimensional integrals may be adapted to this case to show that

${\displaystyle \nabla \cdot \nabla u=0\,}$ in D.

The difficulty with this reasoning is the assumption that the minimizing function u must have two derivatives. Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy. Riemann named this idea the Dirichlet principle in honor of his teacher Peter Gustav Lejeune Dirichlet. However Weierstrass gave an example of a variational problem with no solution: minimize

${\displaystyle W[\varphi ]=\int _{-1}^{1}(x\varphi ')^{2}\,dx\,}$

among all functions φ that satisfy ${\displaystyle \varphi (-1)=-1}$ and ${\displaystyle \varphi (1)=1.}$ ${\displaystyle W}$ can be made arbitrarily small by choosing piecewise linear functions that make a transition between −1 and 1 in a small neighborhood of the origin. However, there is no function that makes ${\displaystyle W=0}$.[lower-alpha 11] Eventually it was shown that Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998).

A more general expression for the potential energy of a membrane is

${\displaystyle V[\varphi ]=\iint _{D}\left[{\frac {1}{2}}\nabla \varphi \cdot \nabla \varphi +f(x,y)\varphi \right]\,dx\,dy\,+\int _{C}\left[{\frac {1}{2}}\sigma (s)\varphi ^{2}+g(s)\varphi \right]\,ds.}$

This corresponds to an external force density ${\displaystyle f(x,y)}$ in D, an external force ${\displaystyle g(s)}$ on the boundary C, and elastic forces with modulus ${\displaystyle \sigma (s)}$ acting on C. The function that minimizes the potential energy with no restriction on its boundary values will be denoted by u. Provided that f and g are continuous, regularity theory implies that the minimizing function u will have two derivatives. In taking the first variation, no boundary condition need be imposed on the increment v. The first variation of ${\displaystyle V[u+\varepsilon v]}$ is given by

${\displaystyle \iint _{D}\left[\nabla u\cdot \nabla v+fv\right]\,dx\,dy+\int _{C}\left[\sigma uv+gv\right]\,ds=0.\,}$

If we apply the divergence theorem, the result is

${\displaystyle \iint _{D}\left[-v\nabla \cdot \nabla u+vf\right]\,dx\,dy+\int _{C}v\left[{\frac {\partial u}{\partial n}}+\sigma u+g\right]\,ds=0.\,}$

If we first set v = 0 on C, the boundary integral vanishes, and we conclude as before that

${\displaystyle -\nabla \cdot \nabla u+f=0\,}$

in D. Then if we allow v to assume arbitrary boundary values, this implies that u must satisfy the boundary condition

${\displaystyle {\frac {\partial u}{\partial n}}+\sigma u+g=0,\,}$

on C. This boundary condition is a consequence of the minimizing property of u: it is not imposed beforehand. Such conditions are called natural boundary conditions.

The preceding reasoning is not valid if ${\displaystyle \sigma }$ vanishes identically on C. In such a case, we could allow a trial function ${\displaystyle \varphi \equiv c}$, where c is a constant. For such a trial function,

${\displaystyle V[c]=c\left[\iint _{D}f\,dx\,dy+\int _{C}g\,ds\right].}$

By appropriate choice of c, V can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless

${\displaystyle \iint _{D}f\,dx\,dy+\int _{C}g\,ds=0.\,}$

This condition implies that net external forces on the system are in equilibrium. If these forces are in equilibrium, then the variational problem has a solution, but it is not unique, since an arbitrary constant may be added. Further details and examples are in Courant and Hilbert (1953).

The Sturm–Liouville eigenvalue problem involves a general quadratic form

${\displaystyle Q[\varphi ]=\int _{x_{1}}^{x_{2}}\left[p(x)\varphi '(x)^{2}+q(x)\varphi (x)^{2}\right]\,dx,\,}$

where ${\displaystyle \varphi }$ is restricted to functions that satisfy the boundary conditions

${\displaystyle \varphi (x_{1})=0,\quad \varphi (x_{2})=0.\,}$

Let R be a normalization integral

${\displaystyle R[\varphi ]=\int _{x_{1}}^{x_{2}}r(x)\varphi (x)^{2}\,dx.\,}$

The functions ${\displaystyle p(x)}$ and ${\displaystyle r(x)}$ are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio Q/R among all φ satisfying the endpoint conditions. It is shown below that the Euler–Lagrange equation for the minimizing u is

${\displaystyle -(pu')'+qu-\lambda ru=0,\,}$

where λ is the quotient

${\displaystyle \lambda ={\frac {Q[u]}{R[u]}}.\,}$

It can be shown (see Gelfand and Fomin 1963) that the minimizing u has two derivatives and satisfies the Euler–Lagrange equation. The associated λ will be denoted by ${\displaystyle \lambda _{1}}$; it is the lowest eigenvalue for this equation and boundary conditions. The associated minimizing function will be denoted by ${\displaystyle u_{1}(x)}$. This variational characterization of eigenvalues leads to the Rayleigh–Ritz method: choose an approximating u as a linear combination of basis functions (for example trigonometric functions) and carry out a finite-dimensional minimization among such linear combinations. This method is often surprisingly accurate.

The next smallest eigenvalue and eigenfunction can be obtained by minimizing Q under the additional constraint

${\displaystyle \int _{x_{1}}^{x_{2}}r(x)u_{1}(x)\varphi (x)\,dx=0.\,}$

This procedure can be extended to obtain the complete sequence of eigenvalues and eigenfunctions for the problem.

The variational problem also applies to more general boundary conditions. Instead of requiring that φ vanish at the endpoints, we may not impose any condition at the endpoints, and set

${\displaystyle Q[\varphi ]=\int _{x_{1}}^{x_{2}}\left[p(x)\varphi '(x)^{2}+q(x)\varphi (x)^{2}\right]\,dx+a_{1}\varphi (x_{1})^{2}+a_{2}\varphi (x_{2})^{2},\,}$

where ${\displaystyle a_{1}}$ and ${\displaystyle a_{2}}$ are arbitrary. If we set ${\displaystyle \varphi =u+\varepsilon v}$ the first variation for the ratio ${\displaystyle Q/R}$ is

${\displaystyle V_{1}={\frac {2}{R[u]}}\left(\int _{x_{1}}^{x_{2}}\left[p(x)u'(x)v'(x)+q(x)u(x)v(x)-\lambda u(x)v(x)\right]\,dx+a_{1}u(x_{1})v(x_{1})+a_{2}u(x_{2})v(x_{2})\right),\,}$

where λ is given by the ratio ${\displaystyle Q[u]/R[u]}$ as previously. After integration by parts,

${\displaystyle {\frac {R[u]}{2}}V_{1}=\int _{x_{1}}^{x_{2}}v(x)\left[-(pu')'+qu-\lambda ru\right]\,dx+v(x_{1})[-p(x_{1})u'(x_{1})+a_{1}u(x_{1})]+v(x_{2})[p(x_{2})u'(x_{2})+a_{2}u(x_{2})].\,}$

If we first require that v vanish at the endpoints, the first variation will vanish for all such v only if

${\displaystyle -(pu')'+qu-\lambda ru=0\quad {\hbox{for}}\quad x_{1}<x<x_{2}.\,}$

If u satisfies this condition, then the first variation will vanish for arbitrary v only if

${\displaystyle -p(x_{1})u'(x_{1})+a_{1}u(x_{1})=0,\quad {\hbox{and}}\quad p(x_{2})u'(x_{2})+a_{2}u(x_{2})=0.\,}$

These latter conditions are the natural boundary conditions for this problem, since they are not imposed on trial functions for the minimization, but are instead a consequence of the minimization.

Eigenvalue problems in higher dimensions are defined in analogy with the one-dimensional case. For example, given a domain D with boundary B in three dimensions we may define

${\displaystyle Q[\varphi ]=\iiint _{D}p(X)\nabla \varphi \cdot \nabla \varphi +q(X)\varphi ^{2}\,dx\,dy\,dz+\iint _{B}\sigma (S)\varphi ^{2}\,dS,\,}$

and

${\displaystyle R[\varphi ]=\iiint _{D}r(X)\varphi (X)^{2}\,dx\,dy\,dz.\,}$

Let u be the function that minimizes the quotient ${\displaystyle Q[\varphi ]/R[\varphi ],}$ with no condition prescribed on the boundary B. The Euler–Lagrange equation satisfied by u is

${\displaystyle -\nabla \cdot (p(X)\nabla u)+q(x)u-\lambda r(x)u=0,\,}$

where

${\displaystyle \lambda ={\frac {Q[u]}{R[u]}}.\,}$

The minimizing u must also satisfy the natural boundary condition

${\displaystyle p(S){\frac {\partial u}{\partial n}}+\sigma (S)u=0,}$

on the boundary B. This result depends upon the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998) for details. Many extensions, including completeness results, asymptotic properties of the eigenvalues and results concerning the nodes of the eigenfunctions are in Courant and Hilbert (1953).

Fermat's principle states that light takes a path that (locally) minimizes the optical length between its endpoints. If the x-coordinate is chosen as the parameter along the path, and ${\displaystyle y=f(x)}$ along the path, then the optical length is given by

${\displaystyle A[f]=\int _{x=x_{0}}^{x_{1}}n(x,f(x)){\sqrt {1+f'(x)^{2}}}dx,\,}$

where the refractive index ${\displaystyle n(x,y)}$ depends upon the material. If we try ${\displaystyle f(x)=f_{0}(x)+\varepsilon f_{1}(x)}$ then the first variation of A (the derivative of A with respect to ε) is

${\displaystyle \delta A[f_{0},f_{1}]=\int _{x=x_{0}}^{x_{1}}\left[{\frac {n(x,f_{0})f_{0}'(x)f_{1}'(x)}{\sqrt {1+f_{0}'(x)^{2}}}}+n_{y}(x,f_{0})f_{1}{\sqrt {1+f_{0}'(x)^{2}}}\right]dx.}$

After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation

${\displaystyle -{\frac {d}{dx}}\left[{\frac {n(x,f_{0})f_{0}'}{\sqrt {1+f_{0}'^{2}}}}\right]+n_{y}(x,f_{0}){\sqrt {1+f_{0}'(x)^{2}}}=0.\,}$

The light rays may be determined by integrating this equation. This formalism is used in the context of Lagrangian optics and Hamiltonian optics.

There is a discontinuity of the refractive index when light enters or leaves a lens. Let

${\displaystyle n(x,y)=n_{(-)}\quad {\hbox{if}}\quad x<0,\,}$

${\displaystyle n(x,y)=n_{(+)}\quad {\hbox{if}}\quad x>0,\,}$

where ${\displaystyle n_{(-)}}$ and ${\displaystyle n_{(+)}}$ are constants. Then the Euler–Lagrange equation holds as before in the region where x<0 or x>0, and in fact the path is a straight line there, since the refractive index is constant. At the x=0, f must be continuous, but f' may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form

${\displaystyle \delta A[f_{0},f_{1}]=f_{1}(0)\left[n_{(-)}{\frac {f_{0}'(0^{-})}{\sqrt {1+f_{0}'(0^{-})^{2}}}}-n_{(+)}{\frac {f_{0}'(0^{+})}{\sqrt {1+f_{0}'(0^{+})^{2}}}}\right].\,}$

The factor multiplying ${\displaystyle n_{(-)}}$ is the sine of angle of the incident ray with the x axis, and the factor multiplying ${\displaystyle n_{(+)}}$ is the sine of angle of the refracted ray with the x axis. Snell's law for refraction requires that these terms be equal. As this calculation demonstrates, Snell's law is equivalent to vanishing of the first variation of the optical path length.

It is expedient to use vector notation: let ${\displaystyle X=(x_{1},x_{2},x_{3}),}$ let t be a parameter, let ${\displaystyle X(t)}$ be the parametric representation of a curve C, and let ${\displaystyle {\dot {X}}(t)}$ be its tangent vector. The optical length of the curve is given by

${\displaystyle A[C]=\int _{t=t_{0}}^{t_{1}}n(X){\sqrt {{\dot {X}}\cdot {\dot {X}}}}\,dt.\,}$

Note that this integral is invariant with respect to changes in the parametric representation of C. The Euler–Lagrange equations for a minimizing curve have the symmetric form

${\displaystyle {\frac {d}{dt}}P={\sqrt {{\dot {X}}\cdot {\dot {X}}}}\,\nabla n,\,}$

where

${\displaystyle P={\frac {n(X){\dot {X}}}{\sqrt {{\dot {X}}\cdot {\dot {X}}}}}.\,}$

It follows from the definition that P satisfies

${\displaystyle P\cdot P=n(X)^{2}.\,}$

Therefore, the integral may also be written as

${\displaystyle A[C]=\int _{t=t_{0}}^{t_{1}}P\cdot {\dot {X}}\,dt.\,}$

This form suggests that if we can find a function ψ whose gradient is given by P, then the integral A is given by the difference of ψ at the endpoints of the interval of integration. Thus the problem of studying the curves that make the integral stationary can be related to the study of the level surfaces of ψ. In order to find such a function, we turn to the wave equation, which governs the propagation of light. This formalism is used in the context of Lagrangian optics and Hamiltonian optics.

The wave equation for an inhomogeneous medium is

${\displaystyle u_{tt}=c^{2}\nabla \cdot \nabla u,\,}$

where c is the velocity, which generally depends upon X. Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy

${\displaystyle \varphi _{t}^{2}=c(X)^{2}\,\nabla \varphi \cdot \nabla \varphi .\,}$

We may look for solutions in the form

${\displaystyle \varphi (t,X)=t-\psi (X).\,}$

In that case, ψ satisfies

${\displaystyle \nabla \psi \cdot \nabla \psi =n^{2},\,}$

where ${\displaystyle n=1/c.}$ According to the theory of first-order partial differential equations, if ${\displaystyle P=\nabla \psi ,}$ then P satisfies

${\displaystyle {\frac {dP}{ds}}=n\,\nabla n,}$

along a system of curves (the light rays) that are given by

${\displaystyle {\frac {dX}{ds}}=P.\,}$

These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification

${\displaystyle {\frac {ds}{dt}}={\frac {\sqrt {{\dot {X}}\cdot {\dot {X}}}}{n}}.\,}$

We conclude that the function ψ is the value of the minimizing integral A as a function of the upper end point. That is, when a family of minimizing curves is constructed, the values of the optical length satisfy the characteristic equation corresponding the wave equation. Hence, solving the associated partial differential equation of first order is equivalent to finding families of solutions of the variational problem. This is the essential content of the Hamilton–Jacobi theory, which applies to more general variational problems.

In classical mechanics, the action, S, is defined as the time integral of the Lagrangian, L. The Lagrangian is the difference of energies,

${\displaystyle L=T-U,\,}$

where T is the kinetic energy of a mechanical system and U its potential energy. Hamilton's principle (or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral

${\displaystyle S=\int _{t=t_{0}}^{t_{1}}L(x,{\dot {x}},t)\,dt}$

is stationary with respect to variations in the path x(t). The Euler–Lagrange equations for this system are known as Lagrange's equations:

${\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {x}}}}={\frac {\partial L}{\partial x}},\,}$

and they are equivalent to Newton's equations of motion (for such systems).

The conjugate momenta P are defined by

${\displaystyle p={\frac {\partial L}{\partial {\dot {x}}}}.\,}$

For example, if

${\displaystyle T={\frac {1}{2}}m{\dot {x}}^{2},\,}$

then

${\displaystyle p=m{\dot {x}}.\,}$

Hamiltonian mechanics results if the conjugate momenta are introduced in place of ${\displaystyle {\dot {x}}}$ by a Legendre transformation of the Lagrangian L into the Hamiltonian H defined by

${\displaystyle H(x,p,t)=p\,{\dot {x}}-L(x,{\dot {x}},t).\,}$

The Hamiltonian is the total energy of the system: H = T + U. Analogy with Fermat's principle suggests that solutions of Lagrange's equations (the particle trajectories) may be described in terms of level surfaces of some function of X. This function is a solution of the Hamilton–Jacobi equation:

${\displaystyle {\frac {\partial \psi }{\partial t}}+H\left(x,{\frac {\partial \psi }{\partial x}},t\right)=0.\,}$

Further applications of the calculus of variations include the following:

• The derivation of the catenary shape
• Solution to Newton's minimal resistance problem
• Solution to the brachistochrone problem
• Solution to isoperimetric problems
• Calculating geodesics
• Finding minimal surfaces and solving Plateau's problem
• Optimal control